Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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31. 0.1, 0.01, 0.001, 0.0001
33. 35.
37.
39.
41.
43.
45.
47. 5 49. 11 51. 9 53. 8 55. 13 57. 9
59. a. appears exponential b. c.
61. a. appears nonexponential b. no common ratio
63. 65. 67.
69. 71. 73. 75. 728
77. 79.
81. 1364
83.
85.
87. 89. 91. 93. 95. no 97.
99. 12 101. 4 103. 105. 107. No finite sum exists.
109. 1296 111.
113.
115.
117. a
0
277; a
n
283.3711.0232
n1
, a
10
⬇ 347.7 million
a
0
160; a
n
155.210.972
n1
, a
8
⬇ 125.4 gpm, 10 mo
a
0
46,000; a
n
36,80010.82
n1
, a
4
$18,841.60, 10 yr
a
n
2410.82
n1
, a
7
⬇ 6.3 ft, S
q
120 ft
3
2
1
3
125
3
27
7
14 151 2
3367
1296
521
25
387
512
31,525
2187
⬇1.60
85
8
10.625
2059
8
257.375
3872
27
⬇ 143.4110,920
r
3
2
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1
256
81
r
3
2
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1
32
243
r
2
3
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1
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200
0
8
50
a
n
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n1
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a
12
0.000008388608a
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0.002048, a
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a
n
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7
16
a
n
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20
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n1
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4
25
4
a
n
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1
2
b
n1
; a
7
3
8
119. 7 strokes
121. 2 days, 8 days
123. bacterium, 12 half-hours (6 hr)
125.
127. about 67,109 in. This is almost 1.06 mi.
129. vs. 40,000(1.04)
x
; 6 yr
131. a. For an arithmetic sequence, the difference must be
constant. For and we have
✓
b. For a geometric sequence, the ratio must be constant.
For and we have
✓
133.
135.
Exercises 9.4, pp. 801–802
1. finite, universally 3. induction, hypothesis 5. Answers will vary.
7.
;
;
9.
;
;
;
11.
13.
15.
S
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2
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15;
S
4
414 12
2
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2
10;
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n
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2
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1k 12151k 12 12 1k 1215k 5 12
S
k
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S
5
515152 12 5125 12 51242 120;
S
4
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S
n
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x 0
10
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40,000 1750x
total distance a
0
2S
q
18 m
a
0
2 m; a
n
1.6 a
4
5
b
n1
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⬇ 0.42 m,
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n
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51,200
a
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8, a
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610.752
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,
a
0
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n
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3
5
b
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⬇ 35.9 in
3
,
SA-46 Student Answer Appendix
cob19545_saa_039-051.qxd 12/17/10 9:29 PM Page SA-46
17.
;
;
;
19. ;
;
;
;
Verified
21. ;
;
;
;
Verified
23. ;
;
;
;
Verified
25. a. ;
b.
27. 1. Show is true for .
Verified
2. Assume is true:
and use it to show the truth of follows. That is:
Working with the left hand side:
Since the truth of follows from , the formula is true for all n.
29. 1. Show is true for .
Verified
2. Assume is true:
and use it to show the truth of follows. That is:S
k1
5 10 15
p
5k
5k1k 12
2
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k
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1
511211 12
2
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2
5
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1
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n 1S
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225
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a
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4
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1
Working with the left hand side:
Since the truth of follows from , the formula is true for all n.
31. 1. Show is true for
Verified
2. Assume is true:
and use it to show the truth of follows. That is:
Working with the left hand side:
Since the truth of follows from , the formula is true for all n.
33. 1. Show is true for
Verified
2. Assume is true:
and use it to show the truth of follows. That is:
Working with the left hand side:
Since the truth of follows from , the formula is true for all n.
35. 1. Show is true for
Verified
2. Assume is true:
and use it to show the truth of follows. That is:
S
k
a
k1
S
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p
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k
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p
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k
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n 1.S
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5 10 15
p
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k
a
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p
5k 51k 12
Student Answer Appendix SA-47
cob19545_saa_039-051.qxd 12/17/10 9:29 PM Page SA-47
Working with the left hand side:
Since the truth of follows from , the formula is true for all n.
37. 1. Show is true for
Verified
2. Assume is true:
and use it to show the truth of follows. That is:
Working with the left hand side:
Since the truth of follows from , the formula is true for all n.
39. 1. Show is true for
Verified
2. Assume is true and use it to show the truth of
follows. That is: .
Working with the left hand side:
Since k is a positive integer,
Showing
Since the truth of follows from P
k
, the statement is true for all n.
41. 1. Show is true for
Verified
2. Assume is true and use it to show the truth of
follows. That is: .
Working with the left hand side:
4
k1
4
414
k
12
4
#
314
k1
2
3
#
4
k
3
#
414
k1
2
3
#
4
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4
k1
1P
k1
P
k
: 3
#
4
k1
4
k
1
3 3
3
#
1 3
3
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4
0
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3
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4
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4
1
1
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1
:
n 1.P
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P
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3
k1
313
k
2
3
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P
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k
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3 2 1
3
1
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1
:
n 1.P
n
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k
S
k1
S
k1
k 1
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12k 121k 12
12k 1212k 32
2k
2
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12k 1212k 32
k
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1
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1
3
1
15
1
35
p
1
12k 1212k 12
1
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S
k
a
k1
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1
121k 12 12121k 12 12
k 1
21k 12 1
1
3
1
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1
35
p
1
12k 1212k 12
S
k1
1
3
1
15
1
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p
1
12k 1212k 12
k
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k
S
1
1
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1
2 1
1
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S
k
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k1
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212
k1
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k1
2 4 8
p
2
k
2
k1
Since k is a positive integer,
Showing :
Since the truth of follows from P
k
, the statement is true for all n.
43. is divisible by 2
1. Show is true for
Verified
2. Assume for and use it to show the truth of
follows. That is: for .
Working with the left hand side:
is divisible by 2.
Since the truth of follows from P
k
, the statement is true for all n.
45. is divisible by 3
1. Show is true for .
Verified
2. Assume for and use it to show the
truth of follows.
That is: for .
Working with the left hand side:
is divisible by 3.
Since the truth of follows from P
k
, the statement is true for all n.
47. is divisible by 5
1. Show is true for .
Verified
2. Assume or for and use it to
show the truth of follows.
That is: for .
Working with the left hand side:
is divisible by 5.
Since the truth of follows from P
k
, the statement is true for all n.
49. verified
51. ;; ;
;;
53.
Mid-Chapter Check, p. 803
1. 3, 10, 17, a
9
59 2. 4, 7, 12, a
9
84 3. 3, 5, a
9
171,
1x 42
2
1y 32
2
25
B
1
c
0.3 0.1
0.4 0.2
dBA c
53
511
dAB c
67
10 0
d
2A 3B c
87
6 7
dA B c
33
1 2
dA B c
11
74
d
P
k1
516m 12
30m 5
30m 6 1
615m 12 1
616
k
2 1
6
k
1
p 僆 ⺪6
k1
1 5p
P
k1
m 僆 ⺪6
k
5m 1P
k
: 6
k
1 5m
1 m
5 5m
6 1 5m
6
1
1 5m
P
1
:
P
n
: 6
n
1 5mn 1P
n
6
n
1
P
k1
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2
2k 12 31k 12
k
3
3k
2
2k 31k
2
2k 12 31k 12
k
3
3k
2
2k 31k
2
2k 12 3k 3
k
3
3k
2
3k 1 31k
2
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3
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p 僆 ⺪1k 12
3
31k 12
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: k
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1 3 2 3m
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1
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P
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2
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k
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2
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p 僆 ⺪1k 12
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m 僆 ⺪P
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:
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n
n
2
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P
k1
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4
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4
k1
4 4
k1
1
SA-48 Student Answer Appendix
cob19545_saa_039-051.qxd 12/17/10 9:29 PM Page SA-48
4. 360 5. 6. d 7. e 8. a 9. b
10. c 11. a.
b.
12. 13. 14.
15. 16. a.
b. 17. 18.
19. 1785 20.
Reinforcing Basic Concepts, pp. 803–804
Exercise 1: $71,500
Exercises 9.5, pp. 811–816
1. experiment, well-defined 3. N
t
5. Answers will vary.
7. a. 16 possible
⬇4.5 ft; ⬇127.9 ft
343
6
n 8, S
8
1640
27
a
1
1
2
, r
1
2
, a
n
1
1
2
2
n
a
1
2, r 3, a
n
2132
n1
S
10
14,762
27
S
10
5n 16, S
16
128n 25, S
25
950
a
1
3
2
, d
3
4
, a
n
3
4
n
3
4
a
1
2, d 3, a
n
3n 1
兺
6
k1
13k 22
b. WW, WX, WY, WZ, XW, XX, XY, XZ, YW, YX, YY, YZ, ZW, ZX, ZY, ZZ
9. 32 11. 15,625 13. 2,704,000 15. a. 59,049 b. 15,120
17. 360 if double veggies are not allowed, 432 if double veggies are allowed.
19. a. 120 b. 625 c. 12 21. 24 23. 4 25. 120 27. 6 29. 720
31. 3024 33. 40,320 35. 6; 3 37. 90 39. 336
41. a. 720 b. 120 c. 24 43. 360 45. 60 47. 60 49. 120
51. 30 53. 60, BANANA 55. 126 57. 56 59. 1 61. verified
63. verified 65. 495 67. 364 69. 252 71. 8! 40,320
73. 8
n
P
r
3 336 75. 20
n
C
r
5 15,504 77. 8
n
C
r
4 70
79. a. b. 81. 7776 83. 324 85. 800
87. 6,272,000,000 89. 518,400 91. 357,696 93. 6720 95. 8
97. 10,080 99. 5040 101. 2880 103. 5005 105. 720
107. 52,650, no 109. 6 · (6
n
C
r
3 2) 2 · 20 148
111. 137; 2415
113.
Exercises 9.6, pp. 823–829
1. 3. 0, 1, 1, 0 5. Answers will vary.
7. 9. S {coach of Patriots, Cougars,
Angels, Sharks, Eagles, Stars}, 11. 13. a. b.
c. d. 15.
17. a. b. 1 c. d. 19. 21. 23. 0.991
25. a. b. c. d. 27. 29. 31. b, about 12%
33. a. 0.3651 b. 0.3651 c. 0.3969 35. 0.9 37. 39. 0.59
41. a. b. c. d. 43. a. b. c. 0 d. e. 1
45. 47. 49. a. b. c. d. e. f.
51. ; answers will vary. 53. a. 0.33 b. 0.67 c. 1 d. 0
e. 0.67 f. 0.08 55. a. b. c. 57. a. b. c.
d. 59. a. b. c. d. e. f. 61. a.
b. c. 63. a. b. c. d. e. 65. a.
b. 67. 69. answers will vary; 20 heads in a row.
71. (9, 1, 1)
73.
x
y
54321
5
4
3
2
1
1
2
3
4
5
2
3
4
5
1
x 僆 31, 02 ´ 31, q2
1
1,048,576
;
1
3360
8
2145
5
429
11
100
9
50
3
100
2
25
47
100
3
16
1
16
1
8
11
26
2
13
9
26
1
13
3
26
3
26
5
16
1
16
1
4
9
16
1
8
1
2
1
2
1
4
;
1
256
5
12
1
36
3
4
8
9
2
9
1
18
11
15
3
4
2
25
9
50
2
25
4
9
1
9
7
36
1
6
7
24
60
143
10
21
5
6
8
9
11
12
1
12
6
7
3
4
1
2
1
4
3
4
P1E
1
2
1
8
, P1E
2
2
5
8
, P1E
3
2
3
4
1
26
1
2
1
4
1
13
P1E2
4
9
1
6
S 5HH, HT, TH, TT6,
1
4
n1E2
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
a
n
1 1n 124;
⬇0.83%⬇1.2%
Exercises 9.7, pp. 835–837
1. one 3. 5. Answers will vary.
7.
9. 11. 13. 35
15. 10 17. 1140 19. 9880 21. 1 23. 1
25.
27.
29.
31. 33. 35.
37. 35x
4
y
3
39. 1792p
2
41. 264x
2
y
10
43. 45. a.
b. 47. a. b. c. d.
49.
51.
53.
Making Connections, p. 837
1. c 3. b 5. d 7. b 9. d 11. a 13. e 15. g
Summary and Concept Review, pp. 838–842
1. 1, 6, 11, 16; 2. 1,
3. 4.
5. 6. 7. 140 8. 35 9. 2, 6, 12, 20, 30 10.
11. 12. a. about 134 hawks b. 8 yr
13. 119 14.
15. 740 16. 1335 17. 630 18. 19. 875 20. 7.55 m
21. 3645 22. 32 23. 2401 24. 10.75 25. 6560 26.
27. does not exist 28. 29. 4 30. 31. does not exist 32. 5
33.
34.
35. (1) Show S
n
is true for ✓
(2) Assume S
k
is true:
Use it to show the truth of :
left hand side:
36. (1) Show S
n
is true for ✓
(2) Assume S
k
is true:
Use it to show the truth of :
1 4 9
p
k
2
1k 12
2
1k 121k 1 12321k 12 14
6
S
k1
1 4 9
p
k
2
k1k 1212k 12
6
n 1: S
1
111 1232112 14
6
1
1k 121k 22
2
k1k 12
2
1k 12
k1k 12 21k 12
2
1 2 3
p
k 1k 12
1 2 3
p
k 1k 12
1k 121k 1 12
2
S
k1
1 2 3
p
k
k1k 12
2
n 1: S
1
111 12
2
1
a
0
1225, a
1
⬇ 1311, a
n
131111.072
n1
, a
15
⬇ 3380, S
15
⬇ 32,944
a
0
121,500, a
1
81,000, a
n
81,000 a
2
3
b
n1
, a
7
⬇ 7111 ft
3
63,050
6561
50
9
819
512
11.25
a
n
3 1221n 12; 65a
n
2 31n 12;
兺
7
i1
1i
2
3i 22; 210
1
2
,
3
4
,
5
4
,
9
4
,
17
4
112
255
256
a
n
17 1n 12132; a
6
2a
n
n
4
; a
6
1296
3
5
,
2
5
,
5
17
; a
10
11
101
a
10
46
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
(2, 0)
(0, 0)
(3, 0)
g1x27 0: x 僆 12, 02 ´ 13, q2
f
132 1
x
y
108642
10
8
6
4
2
2
4
6
8
10
4
6
8
10
2
2
n1
, 2048
⬇61.0%⬇99.0%⬇7.0%⬇0.89%⬇23.0%
⬇17.8%⬇0.25
v
24
6v
22
w
33
2
v
20
w
2
x
9
18x
8
y 144x
7
y
2
11 2i
16x
4
96x
3
216x
2
216x 81
a
6
6a
5
b 15a
4
b
2
20a
3
b
3
15a
2
b
4
6ab
5
b
6
c
5
5c
4
d 10c
3
d
2
10c
2
d
3
5cd
4
d
5
41 38i16x
4
96x
3
216x
2
216x 81
x
5
5x
4
y 10x
3
y
2
10x
2
y
3
5xy
4
y
5
1a 12b22
5
Student Answer Appendix SA-49
WXY
W
Z WXY
X
Z WXY
Y
Z WXY
Z
Z
Begin
cob19545_saa_039-051.qxd 12/17/10 9:30 PM Page SA-49
left hand side:
37. (1) Show P
n
is true for ✓
(2) Assume P
k
is true:
Use it to show the truth of :
left hand side:
Since k is a positive integer, showing
38. (1) Show P
n
is true for ✓
(2) Assume P
k
is true:
Use it to show the truth of :
left hand side:
Since k is a positive integer,
39. (1) Show P
n
is true for ✓
(2) Assume P
k
is true: or for
Use it to show the truth of :
for
left hand side:
is divisible by 2
40. six ways
41. 720; 1000 42. 24 43. 220 44. a. 5040 b. 840 c. 35
45. a. 720 b. 120 c. 24 46. 3360 47. a. 220 b. 1320
48. 49. 50. 51. 52. 53. a. 0.608 b. 0.392
c. 1 d. 0 e. 0.928 f. 0.178 54. a. 21 b. 56
55. a. b.
56. a.
b.
57. a. 280x
4
y
3
b. 58. a. about 93.3% b. about 62.4%
Practice Test, pp. 842–844
1. a. b. 6, 12, 20, 30;
c. 3, 2. a. 165 b. c. d. 7
3. a.
b. c.
d. 4. a. 199 b. 9 c. d. 6
5. a. 1712 b. 2183 c. 2188 d. 12 6. a. b.
7. $6756.57 8. $22,185.27
9.
For : ✓
Assume: is true,S
k
⫽
5k
2
⫺ k
2
S
1
⫽
5112
2
⫺ 1
2
⫽ 2n ⫽ 1
S
k
⫽
5k
2
⫺ k
2
, S
k⫹1
⫽
51k ⫹ 12
2
⫺ 1k ⫹ 12
2
;
a
k
⫽ 5k ⫺ 3, a
k⫹1
⫽ 51k ⫹ 12⫺ 3,
⬇72.4 ft⬇8.82 ft
3
4
a
1
⫽ 10, r ⫽
2
5
, a
n
⫽ 101
2
5
2
n⫺1
a
1
⫽ 4, r ⫽⫺2, a
n
⫽ 41⫺22
n⫺1
a
1
⫽⫺8, d ⫽ 2, a
n
⫽ 2n ⫺ 10
a
1
⫽ 7, d ⫽⫺3, a
n
⫽ 10 ⫺ 3n
⫺2343
512
311
420
212, 17, 16; a
8
⫽ 12
a
8
⫽ 90
1
2
,
4
5
, 1,
8
7
; a
8
⫽
16
11
⫺64,064a
5
b
9
78,125a
7
⫹ 218,750a
6
b ⫹ 262,500a
5
b
2
⫹ 175,000a
4
b
3
a
8
⫹ 813a
7
⫹ 84a
6
⫹ 16813a
5
41 ⫺ 38ix
4
⫺ 4x
3
y ⫹ 6x
2
y
2
⫺ 4xy
3
⫹ y
4
175
396
7
24
5
6
3
13
4
13
A
CB
BC
Begin
BC
CA
AC
BA
AB
⫽ 2q
⫽ 213p ⫹ 12
⫽ 3
#
2p ⫹ 2
⫽ 3
#
2p ⫹ 3 ⫺ 1
⫽ 3
#
12p ⫹ 12⫺ 1
3
k⫹1
⫺ 1 ⫽ 3
#
3
k
⫺ 1
q 僆 ⺪3
k⫹1
⫺ 1 ⫽ 2q
P
k⫹1
p 僆 ⺪3
k
⫽ 2p ⫹ 13
k
⫺ 1 ⫽ 2p
n ⫽ 1: P
1
: 3
1
⫺ 1 ⫽ 2 or 2112
7
k⫹1
⫺ 7 ⱕ 7
k⫹1
⫺ 1.
ⱕ 7
k⫹1
⫺ 7
ⱕ 7
#
7
k
⫺ 1
6
#
7
k
⫽ 7
#
6
#
7
k⫺1
6
#
7
k
ⱕ 7
k⫹1
⫺ 1
P
k⫹1
6
#
7
k⫺1
ⱕ 7
k
⫺ 1
n ⫽ 1: P
1
: 6
#
7
1⫺1
ⱕ 7
1
⫺ 1
4
k⫹1
ⱖ 3k ⫹ 4
12k ⫹ 4 ⱖ 3k ⫹ 4
ⱖ 413k ⫹ 12⫽ 12k ⫹ 4
4
k⫹1
⫽ 414
k
2
4
k⫹1
ⱖ 31k ⫹ 12⫹ 1 ⫽ 3k ⫹ 4
P
k⫹1
4
k
ⱖ 3k ⫹ 1
n ⫽ 1: P
1
: 4
1
ⱖ 3112⫹ 1
⫽
1k ⫹ 1212k
2
⫹ 7k ⫹ 62
6
⫽
1k ⫹ 121k ⫹ 2212k ⫹ 32
6
⫽
k1k ⫹ 1212k ⫹ 12
6
⫹
61k ⫹ 12
2
6
⫽
1k ⫹ 12312k
2
⫹ k ⫹ 6k ⫹ 64
6
1 ⫹ 4 ⫹ 9 ⫹
p
⫹ k
2
⫹ 1k ⫹ 12
2
Prove:
✓
10. For :
✓
Assume:
Prove:
✓
11. a.
b. ABC, ACB, BAC, BCA, CAB, CBA
12. 302,400 13. 14. a. 720 b. 120 c. 20 15. 900,900
16. 302,400 17. a. b.
18. a. b.
19. 0.989 20. a. b. c. d. e. f. g. h. 0
21. a. 0.08 b. 0.92 c. 1 d. 0 e. 0.95 f. 0.03 22. a. 0.1875
b. 0.589 c. 0.4015 d. 0.2945 e. 0.4110 f. 0.2055 23. a.
b. c. d. 24. a. 0.8075 b. 0.0075 c. 0.9925
25. a. about 27.9% b. about 97.6%
Strengthening Core Skills, pp. 845–846
Exercise 1.
Exercise 2.
Exercise 3.
Exercise 4.
Cumulative Review Chapters R–9, pp. 846–847
1. a. 23 cards are assembled each hour. b.
c. 184 cards d.
A
.
M
.
3. ; 5.
7. verified; reflections across
9. a. b.
11.
13. a. b. 15. a. b.
17. (5, 10, 15) 19. ; ( ); ( ), (1, 3);
( ), ( )⫺3 ⫹ 213
, 3⫺3 ⫺ 213, 3
⫺7, 3⫺3, 3
1x ⫹ 32
2
16
⫹
1y ⫺ 32
2
4
⫽ 1
x ⫽ 334x ⬇ 3.19e
5
⫽ 2x ⫺ 1x
3
⫽ 125
x
y
108642
⫺
10
⫺
8
⫺
6
⫺
4
⫺
2
⫺
3
⫺
6
⫺
9
⫺
12
⫺
15
6
9
12
15
3
(⫺2, 0)
(2, 0)
y ⫽ 2
x ⫽ 1x ⫽ ⫺1
(0, 8)
⫺1
1x ⫹ h ⫺ 221x ⫺ 22
4x ⫹ 2h ⫺ 3
y ⫽ x
Y ⫽
kVW
X
; Y ⫽
3VW
2X
x ⬇ 0.91; x ⬇ ⫺2.57x ⫽
⫺5 ⫾ 1109
6
⬇ 6:45
y ⫽ 23x ⫺ 155
⬇ 0.000 388
4
#
10
C
5
52
C
5
4
#
13
C
4
#
39
C
1
52
C
5
⬇ 0.042 917
⬇ 0.326 170
4
#
13
C
3
#
39
C
2
52
C
5
4
C
1
#
13
C
5
⫺ 40
52
C
5
⬇ 0.001 970
47
100
13
100
53
100
59
100
5
12
1
4
7
12
1
2
1
3
5
12
1
4
a
8
⫺ 16a
7
b
3
⫹ 112a
6
b
6
x
10
⫹ 1012 x
9
⫹ 90x
8
⫺4x
4
⫺ 8x
3
y ⫹ 24x
2
y
2
⫺ 32xy
3
⫹ 16y
4
1
3
A
CB
BC
Begin
BC
CA
AC
BA
AB
13
k
⫺ 12
#
3 ⫽ 3
k⫹1
⫺ 3 ⱕ 3
k⫹1
⫺ 1
ⱕ 13
k
⫺ 12
#
3
⫽ 2
#
3
1k⫺12
#
3
2
#
3
1k⫹12⫺1
⫽ 2
#
3
1k⫺12⫹1
2
#
3
1k⫹12⫺1
ⱕ 3
k⫹1
⫺ 1
2
#
3
k⫺1
ⱕ 3
k
⫺ 1
2112ⱕ 2
2
#
3
1⫺1
ⱕ 3
1
⫺ 1n ⫽ 1
15k ⫹ 421k ⫹ 12
2
⫽
15k ⫹ 421k ⫹ 12
2
5k
2
⫹ 9k ⫹ 4
2
⫽
1k ⫹ 1215k ⫹ 42
2
5k
2
⫺ k
2
⫹
101k ⫹ 12⫺ 6
2
⫽
1k ⫹ 12351k ⫹ 12⫺ 14
2
5k
2
⫺ k
2
⫹ 51k ⫹ 12⫺ 3 ⫽
51k ⫹ 12
2
⫺ 1k ⫹ 12
2
SA-50 Student Answer Appendix
cob19545_saa_039-052.qxd 1/8/11 7:15 PM Page SA-50
21. 1333 23. a. b. c.
d. virtually nil
25. a. mid-June
b. c. 98 d. November to February
27.
, (3.4044, 1.4539)
29. 0.2794
CHAPTER APPENDIX
APPENDIX A.I pp. A-10–A-13
1. proper; subset; element 3. positive; negative; 7; ⫺7; principal
5. Order of operations requires multiplication before addition.
7. a. { 1, 2, 3, 4, 5} b. { } 9. True 11. True 13. True
15.
17.
19. a. 40,000 acres
b. ten 3’s (0.333 333 333 3, may vary by calculator model)
21.
23.
25. a. i. { 8, 7, 6} ii. { 8, 7, 6} iii.
iv. v. {}
vi. b.
c.
27. a. i. ii.
iii. iv.
v. vi.
b.
c.
29. False; not all real numbers are irrational.
31. False; not all rational numbers are integers.
33. False; is not irrational.
35. c; IV 37. a; VI 39. d; III
41. Let a represent Kylie’s age: years.
43. Let n represent the number of incorrect words: incorrect.
45. 47. 49. 51. 2.75 53.
55. 57. 59. , 10 61.
63. negative 65. 67. undefined, since implies
69. undefined, since implies k
#
0 ⫽ 77 ⫼ 0 ⫽ k
k
#
0 ⫽ 1212 ⫼ 0 ⫽ k⫺n
⫺8, 2
冟
⫺7.5 ⫺ 2.5
冟
,
冟
2.5 ⫺ 1⫺7.5
冟
3
4
1
2
⫺45 ⱕ w 6 32x 6 2x ⱖ
3
2
n ⱕ 2
a ⱖ 6
125 ⫽ 5
1
⫺
1
⫺
2
⫺
3
⫺
4
⫺
6
⫺
5 2345 760
p兹3 兹49
5⫺5, ⫺3, ⫺1, 0, 13, 2, , 4, 6, 1496
5⫺5, 149, 2, ⫺3, 6, ⫺1, 13, 0, 4, 6513, 6
5⫺5, 149, 2, ⫺3, 6, ⫺1, 0, 465⫺5, 149, 2, ⫺3, 6, ⫺1, 0, 46
5149, 2, 6, 0, 465149, 2, 6, 46
1⫺1⫺223457860
0.75
E
t
5.6
᎑
5⫺1,
3
5
, 0.75,
9
2
, 5.6, 6, 7, 865⫺1, 8, 0.75,
9
2
, 5.6, 7,
3
5
, 66
5⫺1, 8, 0.75,
9
2
, 5.6, 7,
3
5
, 66
1⫺1, 8, 7, 62
234501
⬇1.73
0123⫺2 ⫺1
⬇2.65
12340⫺1
2.5
12340⫺1
1.3
20
0
10
0
⬇142
y ⫽⫺7x
2
⫹ 91x ⫺ 154•
9a ⫹ 3b ⫹ c ⫽ 56
25a ⫹ 5b ⫹ c ⫽ 126
81a ⫹ 9b ⫹ c ⫽ 98
;
a
12
0
b 10.042
12
10.962
0
;
⬇98.9%⬇91.9%⬇7.0%
71. a. positive b. negative c. negative d. negative
73. 75. 77. is closest 79. 7 81. 83.
85. or 87. 0 89. 91. 93. 95. 97.
99. 64 101. 4489.70 103. 3 105. 107.
109. 111. about 68.13 cm 113. Tsu Ch’ung-chih:
115. negative
APPENDIX A.II pp. A-22–A-27
1. composite 3. space 5. Answers will vary.
7. or 22.8 cm
2
9.
11. 13. 15.
17. 19.
21. 23.
25.
27. 504 in
2
29.
31. a.
b. Total Area area of larger rectangle plus area of smaller rectangle
c.
33. a.
b. Total volume ⫽ volume of rectangular solid plus volume of pyramid
c.
35.
37. 156.25 gal 39. 12.5 gal 41. 7108.4 yd 43. 661.5 kg
45. K: LW C: J:
A: 4S I: LWH D:
H: E: G: S
2
B: F: S
3
L:
47. 32 ft 49. 192 ft
3
51. 23.91 m
53. 55. one large at about
57. about 654.5 gal 59. a. 35.3 m
2
b. 2.1 m
3
c. about $280.88
61. a. 216.5 ft
2
b. about 7 c. about $61.07 63.
65.
about 58,752 gal
67. d. 35 ⫹ xy ⫺ 7y
2500 ⬇ 7854 ft
3
5 in. ⫻ 3 in.
5.8¢/in
2
91.44 cm ⫻ 76.2 cm ⫻ 121.92 cm
h1b
1
⫹ b
2
2
2
2L ⫹ 2W
2rS
1
⫹ S
2
⫹ S
3
⫹ S
4
1
2
BH
r
2
S
1
⫹ S
2
⫹ S
3
V ⫽ 200 ⫹ 43.75 ⬇ 337.4 in
3
V ⫽ LWH ⫹ r
2
h
V ⫽ LWH ⫹ LWh or V ⫽ LWH ⫹ s
2
h
W
H
L
h
s
s
A ⫽ LW ⫹ Iw
⫽
w
l
L
W
W ⫽ 12 cm
A ⫽ 9600 ⫹ 1600 ⬇ 14,626.5 ft
2
A ⫽ LW ⫹ r
2
P ⫽ 240 ⫹ 80 ⬇ 491.3 ft;
P ⫽ 2L ⫹ d
A ⫽ 81 ⬇ 254.5 in
2
A ⫽ 54 ft
2
A ⫽
r
2
4
A ⫽
bh
2
⫹ LW
P ⫽ 108 cm
A ⫽ 486 cm
2
P ⫽ 21d2 15 ⬇ 47.1 in.c ⫽ 45 cm
A ⫽ 24 cm
2
A ⫽ 20 cm
2
C ⬇ 15.7 cmP ⫽ 24 cmP ⫽ 18 cm
A ⬇ 950 mm
2
A ⫽ 2280 mm
2
355
113
179°F
32°FD ⬇ 4.3 cm
⫺11
12
⫺4⫺
7
8
⫺
1
10
⫺5⫺2
5
12
⫺
29
12
4
1
3
⫺2.1859
2
⫽ 81⫺2⫺
11
6
Student Answer Appendix SA-51
cob19545_saa_039-052.qxd 12/23/10 9:06 PM Page SA-51
This page intentionally left blank
Index
I-1
A
Absolute maximum, 196
Absolute value
of complex numbers, 291, 408
explanation of, 81, 220, 257
on graphing calculator, 221, 224, 225, A–6
multiplicative property of, 222
of real numbers, A–5 – A–6
Absolute value equations
distance and, 231
explanation of, 275
graphically solving, 226
on graphing calculator, 221
method to solve, 220–222, 275
property of, 220
Absolute value functions, 204, 225, 257
Absolute value inequalities
applications involving, 226–227
distance and, 231
explanation of, 275
graphically solving, 226
on graphing calculators, 226
solutions to “greater than,” 224–225
solutions to “less than,” 223–224
Accumulated value, 540, 543
Addition
associative property of, 5, 81
commutative property of, 5, 81
of complex numbers, 284–285
distributive property of multiplication over,
6–7, 285, 768
of matrices, 652–653, 700
of polynomials, 17–18
of radical expressions, 70
of rational expressions, 56–58
Addition method. See Elimination
Addition of ordinates, 351
Additive identity, 5–6, 81
Additive inverse, 6, 81
Additive property
of equality, 25
of inequality, 27–28
Algebra
of functions, 340–346, 373
fundamental theorem of, 394–397, 399
of matrices, 650–658
Algebraic expressions
evaluation of, 3–4
explanation of, 2
method to simplify, 7
translating English phrases into, 3
Algebraic fractions. See Rational expressions
Algebraic method
explanation of, 482
to find inverse functions, 483–485
Algebraic terms, 2
Algorithm, division, 383
Allometric studies, 241
Amortization, 542–545
Analytical geometry
algebraic tools used in, 708–709
conic section characteristics and, 710–712
distance between point and line and, 709
explanation of, 708, 755
Angles, right, A–14
Annuities, 542–545
Applications. See also Applications index
of absolute value inequalities, 226–227
of algebra of functions, 345–346
of analytic parabolas, 750
of binomial theorem, 834–835
of combinations, 810–811
of composition, 362–363
of data analysis, 170
of difference quotient, 362–364
of direct variation, 262–264
of exponential and logarithmic functions,
492, 498–499, 508–512, 534–535,
539–547
of foci of conic sections, 724–725,
739–740
of functions, 189
of inequalities, 32, 464–465
of inverse functions, 487
of inverse variation, 265
of joint variation, 266
of linear equations, 31–32, 113–114,
142–144
of linear inequalities, 615
of linear programming, 622–625
of matrices, 645–646, 672–673
of piecewise-defined functions, 254–255
of polynomial functions, 405, 421–422
of power functions, 239–242
problem-solving guide for, 157–160
of quadratic equations, 306–307
of quadratic inequalities, 307–308
of rate of change, 332–334, 362–364
of rational functions, 239–242, 439,
451–454
of regression, 556–557
of remainder theorem, 390
of sequences, 768–769, 779, 789–791
of series, 790–791
of system of linear equations in three
variables, 598–599
of system of linear equations in two
variables, 582–586
of system of linear inequalities, 619–620
Area. See also Surface area
explanation of, 32, A–14
formulas for, 33, 37, 455, 491, 627, 660,
727, A–14 – A–16
of triangle, 655, 688
Arithmetic sequences
applications of, 779
explanation of, 773, 839
finding nth partial sum in, 777–778
finding nth term of, 774–777
graphs of, 776
identification of, 773–774
Associated minor matrices, 669–670
Associations
linear/nonlinear, 166–167
positive and negative, 165
Associative properties, 5, 81
Asymptotes
of central hyperbola, A–33
to graph rational functions, 235–236
horizontal, 234, 240, 433–435
oblique or nonlinear, 448–450
vertical, 235, 240, 430–433
Asymptotic behavior, 234, 733
Augmented matrices
explanation of, 700
matrix inverses and, 704–705
of system of equations, 638–640, A–30
triangularizating, 640–642
Auxiliary lines, A–15
Average distance, 95
Average rate of change
applications of, 333, 334, 362–364
calculation of, 334
difference quotient and, 359–361
explanation of, 332, 373
Average rate of change formula, 361
Axis of symmetry, 205
B
Back-substitution, 26, 578
Base, A–7, A–21
Base-b, to solve exponential equations, 530
Base-e exponential functions, 495–496
Binomial coefficients, 831–832
Binomial conjugates, 19–20
Binomial cubes, 291
Binomial expansion, 830, 831, 833, 834
Binomial experiment, 834
Binomial factors, 40–41
Binomial powers, 829–831
Binomial probability, 834–836
Binomials. See also Polynomials
expansion of, 830, 831, 833, 834
explanation of, 16, 829
multiplication of, 18–19, 285
Binomial squares, 20–21, 43, 759, 830
Binomial theorem
applications of, 834–835
binomial expansion and, 834
explanation of, 829, 833, 842
Bisection, 429
Body mass index formula, 37
Boundary lines
horizontal, 123–124, 134
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I-2 Index
Boundary lines—Cont.
linear inequalities and, 615–616
vertical, 122–123, 134
Boundary point, 122
Bounded region, 621
Branches, of hyperbola, 731
Break-even analysis, 584, 610
C
Calculators. See Graphing calculators
Capacity, A–18 – A–19
Cardano, Girolomo, 291
Carrying capacity, 555
Center
of circle, 96–98
of hyperbola, 732, 733
Centimeters of mercury, 510
Central circle, 96, 718
Central ellipse, 718
Central hyperbola, 731–732, A–33
Central rectangle, 733
Change in x, 107
Change in y, 107
Change-of-base formula, 522–523
Circles
center of, 96–98
central, 96, 718
circumference of, 176, 261, 709, A–15
equation of, 96–97, 715–716, 755
explanation of, 33, 96, 710, 755
on graphing calculator, 99–100, 715
graphs of, 95–99, 715–716
perimeter and area formulas for, 33, A–14
properties of, 714
radius of, 96–98, 103, 491, 710, 714
standard form of, 96
Circumference, 176, 261, 709, A–15
Clark’s rule, A–12
Coefficient matrices, 638–639, 700
Coefficients
explanation of, 2
leading, 17, 41
solving linear equations with fractional, 26
Coincident dependence, 598
Collinear points test, 688
Combinations
explanation of, 809–810
on graphing calculator, 810
stating probability using, 820
Combined variation, 266
Common difference, for sequence, 773
Common logarithms
explanation of, 504
method to find, 505–506
Common ratio, 782
Commutative properties, 5, 81
Complementary events, 818–819
Complements, 818–820
Completing the square
applications of, 730–731, 759
explanation of, 104, 297
to graph circles, 715, 716
to graph ellipses, 719–720, 722–723
to graph hyperbolas, 735–736, 738
to graph parabolas, 746
to solve quadratic equations, 297–299
to solve quadratic functions, 313–315
Complex conjugates theorem
division and, 288
explanation of, 286, 396
product of, 286
proof of, A–34 – A–35
Complex numbers
absolute value of, 291, 408
addition and subtraction of, 284–285
division of, 288
explanation of, 283, 370–371
on graphing calculator, 285, 286
identifying and simplifying, 282–284
multiplication of, 285–287
raised to power, 831
square root of, 408
standard form of, 283–284
Complex polynomial functions, 394
Complex polynomials, 312
Complex zeroes, 475–476
Composite figures
explanation of, A–15 – A–16
volume of, A–17 – A–18
Composite functions, 356–357
Composition applications, 362–363
Composition of functions
explanation of, 352–354, 374
on graphing calculator, 355, 358–359
method to find, 354–356
notation for, 353
numerical and graphical view of,
357–359
Compounded interest formula, 541
Compound fractions, 58–59
Compound inequalities
explanation of, 29
method to solve, 30–31
Compound interest
explanation of, 540–542, 569
on graphing calculator, 542
Compressions, 210–211
Conditional equations, 27
Cones
explanation of, 710
surface area of, 80
volume of, 35
Conical shells, 52
Conic sections. See also Circles; Ellipses;
Hyperbolas; Parabolas; specific conic
sections
applications of foci of, 724–725, 739–740
characteristics of, 710–712
equations of, 736–737, A–32 – A33
explanation of, 708
graphs of, 200
nonlinear systems and, 749–750
Conjugate axis, 732
Conjugates
binomial, 19–20
complex, 286, 288, 396
Consecutive integers, 158
Constant of variation, 261
Constant terms, 2
Constraints, 620
Continuous functions
explanation of, 137
piecewise and, 247, 251
Continuous graphs, 91
Continuously compounded interest, 541–542,
545, 569
Contradictions, 27
Convenient values method, 685–687
Convex region, 621
Coordinate, A–2
Coordinate grid, 89
Correlation, strong and week, 167–169
Cost-based pricing, 584
Counting techniques
combinations, 809–811
distinguishable permutations, 807–808
fundamental principle of, 806–807
listing and tree diagrams, 804–806
nondistinguishable permutations, 809
review of, 840–841
Cramer’s Rule, 680–682, 701, 704
Cube root function, 205
Cube roots, 66, A–8. See also Radical
expressions
Cubes
binomial, 291
explanation of, 35, A–21
perfect, 44–45, 394, A–7
sum of cubes of n natural numbers, 793
sum or difference of two perfect, 44–45
volume of, A–17
Cubic units, A–16
Cubing function, 205
Cylinders, 52, 162, 176, 311, 349, 455
Cylindrical shells, 52
D
Data analysis, applications of, 170
Decay rate, 545–547
Decimal notation, 15–16
Decimals, A–2, A–3
Decision variables, 622
Decomposition
of composite function, 356–357
for rational expressions, 682,
685–688
of rational terms, 2
Decomposition template, 682–685
Degenerate cases, 104
Degree, of polynomials, 16
Delta (⌬), 107
Denominator, rationalizing the, 71
Dependent system of equations
matrices and, 644–645
in three variables, 596–598
in two variables, 581–582
Dependent variable, 88
Depreciation, 498
Descartes’rule of signs, 402–403
Descriptive variables, 2
Determinants
to find area of triangle, 688
of general matrices, A–31
of singular matrices, 668–672
to solve systems, 679–682
3 ⫻ 3, 670–671
2 ⫻ 2, 701
Diagonal entries, of matrices, 638
Difference of two squares, factoring, 43
Difference quotient
applications of, 362–364
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Index I-3
average rate of change and, 359–361
explanation of, 337, 359, 374
Directrix
of ellipse, 711, 712
of hyperbola, 712
of parabola, 710, 711, 746–749
Direct variation
applications of, 262–264
explanation of, 261, 277
Discontinuities
asymptotic, 431
removable, 446–447
Discontinuous functions, piecewise-defined,
252, 253
Discriminant
of cubic equation, 467
explanation of, 301
of quadratic formula, 301–302
use of, 312
Diseconomies of scale, 610
Distance
absolute value and, 231
average, 95
perpendicular, 601, 709
Distance formula, 95–96, 708, 755
Distinguishable permutations, 807–808
Distributive property of multiplication over
addition, 6–7, 285, 768
Division
of complex numbers, 288
factor theorem and, 387–389
on graphing calculator, A–7
long, 382
with nonlinear divisor, 385
of polynomials, 383, 384
of radical expressions, 71–72
of rational expressions, 56
remainder theorem and, 386–387, 390
synthetic, 383–384, 470, A–28 – A–29
with zero, 384–385, A–6
Division algorithm, 383
Divisor, 385
Domain
of functions, 123–125, 128
implied, 124–125
of logarithmic functions, 507–508
of piecewise-defined functions, 248–249
of power function, 237–238
of rational functions, 431
of relations, 88, 134–135
solving quadratic inequality to determine,
305–306
Dominant term, of polynomial function, 413
E
e, 495–496
Eccentricity, 758–759
Economies of scale, 610
Electrical resistance, 10
Elementary row operations, 640–641
Elements of set, A–1
Elimination
explanation of, 579, 580, 634–635
Gaussian, 40, 642, A–30
Gauss-Jordan, 642–645, A–30
to solve system of linear equations in three
variables, 593–596
to solve system of linear equations in two
variables, 579–581
to solve system of nonlinear equations,
607–608
Ellipses
applications using characteristics of,
724–725
area of, 727
central, 718
characteristics of, 711–712, 724–725
definition of, 720, 721
equation of, 716–717, 720–721, 723–724,
755, A–32
focal chord, 723
foci of, 720–725
graphs of, 717–724
horizontal, 717–718
perimeter of, 727
with rational/irrational values, 759
vertical, 718–719
Elongation, 758–759
Empty set, A–1
End-behavior
explanation of, 195
of polynomial graphs, 412–415, 419
of rational functions, 233, 234
Endpoint maximum, 196
Endpoints, 122, A–4
Equality
additive property of, 25
of matrices, 650–651
multiplicative property of, 25
power property of, 72, 74
square root property of, 295–297
Equations. See also Linear equations; System
of linear equations; System of linear
equations in three variables; System
of linear equations in two variables;
System of nonlinear equations;
specific types of equations
absolute value, 220–222, 226, 231
of circle, 96–97, 715–716, 755
conditional, 27
of conic sections, 736–737
of ellipse, 716–717, 720–721, 723–724,
755, A–32
equivalent, 25
explanation of, 25, 150
exponential, 496–499, 503–505, 517–519,
527, 531–533, 569
families of, 25
in function form, 137, 138
of functions, 236, 249, 253–254,
316, 438
of hyperbolas, 731–736, 756,
A–32 – A–33
intersection-of-graphs method to solve,
150–153, 183, 519
of line, 138–141
literal, 155–156
logarithmic, 517–519, 527–530, 569
logistic, 533–534, 572
matrix, 663, 667–668
of parabola, 746–749, 756
parametric, 467
of piecewise-defined function, 249,
253–254
polynomial, 46–48, 312
present value, 540
quadratic, 46–48, 292–302, 371
of quadratic function, 316
radical, 72–76
rational, 59–60
of rational function, 236, 438
regression, 171–172, 552, 556–557, 570
relations stated as, 89
roots of, 25
of semi-hyperbola, 742
variation, 261, 263, 265
written information translated into,
31–32
Equivalent equations, 25
Equivalent system of equations, 579,
593–594
Euler, Leonhard, 37
Euler’s polyhedron formula, 37
Even functions, 190–191
Events
complementary, 818–819
explanation of, 816
mutually exclusive, 822
nonexclusive, 821–823
probability of, 817, 818
Experiments, 805
Exponential decay, 545–547
Exponential equations
explanation of, 569
logarithmic form and, 503–505
method to solve, 517–519, 531–533
uniqueness property to solve, 496–499
Exponential form, 504–505, 519, A–7
Exponential functions. See also
One-to-one functions
applications of, 492, 498–499
base-e, 495–496
evaluation of, 492–493
explanation of, 492, 566–567
on graphing calculator, 496–499
graphs of, 493–495
natural, 496
Exponential growth, 495, 545–546
Exponential notation, 11, A–7
Exponential properties
multiplying terms using, 11
simplifying expressions using, 14
summary of, 15
Exponential regression model, 553–554
Exponents
explanation of, 11, A–7
power property of, 11–12, 81
product property of, 11, 12, 81
quotient property of, 13, 81
rational, 67–68
zero and negative, 13, 81
Extraneous roots, 60, 527
Extrapolation, 556–557, 570
Extreme values
explanation of, 316
quadratic functions and, 316–321
F
Factorial formulas, 814
Factorial notation, 765
Factorials, 764–765
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